I have a feeling my answer is incorrect, but it might give you an idea on how to approach

Let m = p^x * t ^ y

Question is basically asking is m/(p^2)t an integer.

Rephrasing, this means is p^(x-2) * t ^ (y-1) an integer

(1)

Total number of factors of m > 9
=> (x+1)(y+1) > 9

A variety of values of x and y satisfy this equation. E.g., if x = 3, y =4
then our equation asks is p^(3-2)*t(4-1) an integer = p*t^3 which is an integer

However, if x = 1, y = 11
p^(-1)*t^10 = t^10 / p which is not an integer

Hence insufficient

(2)

Let m = p^3 * K, where K is some constant

Question is now asking:

Is [(p^3) * K] / [(p^2) * t] an integer
= p*K/t

This may or may not be an integer, depending on the value of K. Hence, insufficient.

Combining:

m = p^x * t^y = p^3 * k

=> x = 3
We also know that (x+1)(y+1) > 9, this means 4(y+1)>9 => y + 1 > 2
The minimum value of y + 1 has to be 3 => min. value of y = 2

p^(x-2) * t^(y-1) will thus always be an integer, since x = 3 and minimum value of y is 2

Thus (C)

Can you please confirm the OA?

Answer is B

ruchisharma wrote:If the prime numbers, p & t are the only prime factors of the integer m, is m a multiple of (p^2)t?

1.) m has more than 9 positive factors.

2.) m is a multiple of p^3

the question is basically asking whether m is divisible by (p^2)t - which would mean whether we get an integer or not

stmt (1)
clearly insufficient. It does not tell us anything about P or T or "prime" factors.

Stmt (2)

if m/P^3 is an integer. clearly, m/p^2 would be an integer. And since T is a prime factor of M, then the result of m/P^2 multiplied by T would also be a factor of M.

this one is like saying if X,Y are factors of Z, then XY would also be factors of Z. This is like going up the prime factorization tree.

Therefore the answer is (B).

ruchisharma wrote:If the prime numbers, p & t are the only prime factors of the integer m, is m a multiple of (p^2)t?

1.) m has more than 9 positive factors.

2.) m is a multiple of p^3

Asking: if p, p and t, are factors of m.
1) is insufficient because m having more than 9 positive factors doesnt tell us if it has p and p as prime factors.
2) is sufficient because it tells you that p, p and p are factors of m..